Chapter 4.4, Problem 36PPS

To determine

To find: The coordinates of the image after transformation.

Answer to Problem 36PPS

The new coordinate of the image after transformation are $A\left(1,-2\right),B\left(2,-4\right),C\left(5,-1\right)$ .

Explanation of Solution

Given:

The given image is shown in Figure 1

  

Figure 1

The image is reflected in $y=x$ and then rotated $270°$ about the origin.

Calculation:

Consider the vertices of the image are,

  $A\left(1,2\right),B\left(2,4\right),C\left(5,1\right)$

The vertices in the matrix form are,

  $\left[\begin{array}{ccc}1& 2& 5\\ 2& 4& 1\end{array}\right]$

Consider the transformation matrix is,

  $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Then, the vertices after transformation is,

  $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}1& 2& 5\\ 2& 4& 1\end{array}\right]=\left[\begin{array}{ccc}1& 2& 5\\ -2& -4& -1\end{array}\right]$

The graph for the coordinates after transformation is shown in Figure 1

  

Figure 1

Then, the coordinates of the transformed image is,

  $A\left(1,-2\right),B\left(2,-4\right),C\left(5,-1\right)$